Limite successione mediante il criterio del confronto asintotico

Si prova che $\forall \ n\geq 1$ vale

$$e-\left (1+\frac{1}{n}\right )^n\geq\frac{e}{2n+2}$$

Pertanto

$$n^2\left [e-\left (1+\frac{1}{n}\right )^n\right ]\geq n^2\frac{e}{2n+2}\stackrel{n\to +\infty}{\longrightarrow} +\infty$$

e

$$n\left [e-\left (1+\frac{1}{n}\right )^n\right ]\geq n\frac{e}{2n+2}\stackrel{n\to +\infty}{\longrightarrow} \frac{e}{2}$$